Finely chop onions and mince garlic.
In a 6-quart heavy saucepan combine onions, garlic, vinegar, butter, ketchup, Worcestershire sauce, 2 tablespoons Tabasco, salt, and pepper and simmer, covered, stirring occasionally, 15 minutes.
Add turkey breast, cavity side down, and cook at a bare simmer, covered, 2 1/2 hours.
Transfer breast to a cutting board, reserving sauce, and cool until it can be handled. Shred turkey meat, discarding bones, and stir meat into reserved sauce.
Simmer barbecue, covered, over low heat, stirring occasionally, 1 1/2 hours more and season with salt, pepper, and additional Tabasco.
Serve barbecue on rolls.